JAMING, Philippe; KELLAY, Karim; PEREZ, Rolando

doi:10.1007/s00605-022-01716-y

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JAMING, Philippe
Institut de Mathématiques de Bordeaux [IMB]

KELLAY, Karim
Institut de Mathématiques de Bordeaux [IMB]

PEREZ, Rolando
Institut de Mathématiques de Bordeaux [IMB]
Institute of Mathematics, University of the Philippines Diliman

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Ce document a été publié dans

Monatshefte für Mathematik. 2022-08, vol. 198, n° 4, p. 757-776

Springer Verlag

Résumé en anglais

In the classical phase retrieval problem in the Paley-Wiener class $PW_L$ for $L>0$, \textit{i.e.} to recover $f\in PW_L$ from $|f|$, Akutowicz, Walther, and Hofstetter independently showed that all such solutions can be obtained by flipping an arbitrary set of complex zeros across the real line. This operation is called zero-flipping and we denote by $\mathfrak{F}_a f$ the resulting function. The operator $\mathfrak{F}_a$ is defined even if $a$ is not a genuine zero of $f$, that is if we make an error on the location of the zero. Our main goal is to investigate the effect of $\mathfrak{F}_a$. We show that $\mathfrak{F}_af$ is no longer bandlimited but is still wide-banded. We then investigate the effect of $\mathfrak{F}_a$ on the stability of phase retrieval by estimating the quantity $\inf_{|c|=1}\|cf-\mathfrak{F}_af\|_2$. We show that this quantity is in general not well-suited to investigate stability, and so we introduce the quantity $\inf_{|c|=1}\|c\mathfrak{F}_bf-\mathfrak{F}_af\|_2$. We show that this quantity is dominated by the distance between $a$ and $b$.< Réduire

Mots clés en anglais

phase retrieval

Paley-Wiener class

zero-flipping

stability

URI

https://oskar-bordeaux.fr/handle/20.500.12278/191602

DOI

http://dx.doi.org/10.1007/s00605-022-01716-y

Project ANR

Noyaux reproduisants en Analyse et au-delà - ANR-18-CE40-0035

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Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251